On inverse problem for singular Sturm-Liouville operator with discontinuity conditions (Q464549)

The paper deals with the discontinuous boundary value problem \(L\) for the Sturm-Liouville equation with a Coulomb singularity \[ -y''+\frac{C}{x}y+q(x)y=\lambda y, \quad 0<x<\pi, \] under the Dirichlet boundary conditions \[ y(0)=y(\pi)=0 \] and the spectral parameter dependent discontinuity conditions \[ y(d+0)-\alpha y(d-0)=y'(d+0)-\frac1\alpha y'(d-0)-2k\beta y(d-0), \] where \(C,\alpha,\beta\in{\mathbb R},\) \(\alpha\neq1,\) \(\alpha>0,\) \(d\in(\pi/2,\pi),\) \(q(x)\) is a real-valued function from \(L_2(0,\pi)\) and \(\lambda=k^2\) is the spectral parameter. The authors study spectral properties of \(L\) and obtain the completeness and expansion theorem. They also prove the uniqueness of recovering \(L\) from three different pieces of data: the Weyl function, the spectral data consisting of the spectrum with the norming constants, and two spectra.